Many acceptor and grammar systems are known to be semilinear full trios, such as finite-index
\(\)  systems [1]}, indexed grammars with a bound on the number of variables appearing in every sentential form (called uncontrolled finite-index) [2]}, multi-push-down machines (which have \(k\)  pushdowns that can simultaneously be written to, but they can only pop from the first non-empty pushdown) [3]}, a Turing machine variant with one finite-crossing worktape [4]}, and pushdown machines that can flip their pushdown up to \(k\)  times [5]}.

Let \({\cal L}\)  be any of the following families:
